Presented By: Neha Sisodiya (201111032)

1. Robust Histogram Construction from Color Invariants for ObjectRecognition Authors : Theo Gevers and Harro Stokman Presented By: Neha Sisodiya (201111032)

2. Introduction • Color is a useful cue for object identification. • An object's color is not an intrinsic property of the object itself. • Depends on confoundingimaging condition (e.g., illumination, shading, highlights, and interreflections) • Color invariants :Transformations of ``color'' which are invariant • Normalized color rgb (c1c2c3) color model • Hue color space (l1,l2,l3) color model

3. Image Formation Figure : A simple model of image formation: light from a source (e.g. the sun) is incident upon and reflected from a surface. The reflected light enters the eye leading eventually to our perception of colour.

4. The response R,G,B is given by Qx(λ) is characterize theresponse of incident light to color signal C(λ) is the color signal product of the light incident upon a surface E(λ) and surface reflectance function S(λ) • the intensity of the reflected light depends on the cosine of the angle between the surface normal (a vector denoted ) and the angle of incidence of the light (a vector denoted )

5. The equation above surface/lighting geometry changes, sensor responses change by the same scale factor • It is thus straightforward to remove the dependence of sensor responses on lighting geometry. • So we can have rgb co-ordinates as below referred to as chromaticity co-ordinates . • invariant to a change in camera viewpoint, object pose, and the direction and intensity of the incident light. • But normalized color rgb is is unstable (or undefined at the black point (R = G = B = 0)

6. Hue, Saturation ,and Value (HSV) • Hue refers to a pure color i.e linear mixtures between adjacent pairs • Model is cylindrical rearrange the geometry of RGB in an attempt to be more intuitive and perceptually relevant than the cartesian (cube) representation . • Saturation can be defined as either the ratio of colorfulness to brightness • The hue θ is below,

7. Effect of Sensor Noise • rgb and H(hue) become very unstable at singularities. • Traditionally is suppressed by ad hoc thresholding of the transformed values prior to color histogram construction for object identification • Proposed technique is to estimate the noise propagation through color invariants. • They use VariableKernel Density estimators to suppress the effect of noise to construct color invariant histograms. • Associated uncertainty is known for each color invariant value and this is used to is used to derive the optimal parameterization of the variable kernel used during histogram construction.

8. Photometric Color Invariance • The reflection from inhomogeneous dielectric materials under white, spectrally smooth illumination is given by , k є(R,G,B), Fk(λ) are spectral sensitivities , B(λ) is surface albedo E denotes the white light source and S is the Fresnel reflectance. • Normalized color rg is color invariant by substituting r and g in (1) • Above equation only dependent on the sensors and the surface albedo. • The opponent color space defined by

9. The opponent color space o1o2 is independent of highlights as follows from substituting (1) in (5) and (6) • Hue can be computed by substituting the reflection term of (1) in θ we will get

10. Noise Propagation • Additive Gaussian noise is widely used. • Assumption sensor noise is normally distributed. • True value of a measurandu is related to its N arguments, denoted by uj as follows • We estimate û of u we will get û1,.,.,.,.,.,.,.,...ûNmeasured with standard deviation

11. Approximation by taylor series w.r.t noise given by where û1=u1+Є1 and û2=u2+Є2 • The error of an indirect measurement is in general • In general, only the first linear term is used to compute the error

12. If the uncertainties inû1,.,.,.,.,.,.,.,...û N are independent, random, and relatively small, the predicted uncertainty in q is given by eq(16) • Substitution of (2) and (3) in (16) gives the uncertainty for the normalized coordinates • The uncertainties of o1 and o2 are given by eq(18)

13. Substitution of (8) in (16) gives the uncertainty for the hue

14. HISTOGRAM CONSTRUCTION BY VARIABLE KERNEL DENSITY ESTIMATION • The (one-dimensional) histogram is defined as eq(20) • Problem is Bin-width parameter and the position of the bin edges to be choosen • The kernel density estimator is insensitive to the placement of the bin edges

15. Variable kernel density estimator eq(22) • Assuming normally distributed noise, the distribution is approximated well by the Gauss distribution • the variable kernel method estimating the univariate,directional hue density is as follows eq(24) where σθ is derived according to (19), with corresponding value θ.

16. The variable kernel method for the bivariate normalized rg kernel is given by where σr,σgare derived according to (17). • Similarly bivariate normalized o1 o2 kernel is given by where σo1,σo2are derived according to (18).

17. Experimental Results

18. Experimental Results ….. Fig. 2. Various images which are included in the image database of 500 images. The images are representative for the images in the database. Objects were recorded in isolation (one per image).

19. Robustness Against Noise: Simulated Data Fig. 3. Two images generating together 10 images by adding noise with   (8,16,32,64,128)

21. Robustness Against Noise: Realistic Data Fig. 6. Two objects under varying illumination intensity generating each four images with SNR (24,12,6,3).

22. Comments • Kernel density estimator overcome the problem of ad hoc thresholding at unstable color invariants. • The variable kernel density estimation is derived from a theoretical framework for noise propagation through color invariants. • Proposed method is less sensitive to Gaussian noise than traditional histogram construction schemes. • It is computationally more expensive.

23. Thanks